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Properties

Label 2-1920-80.3-c1-0-45
Degree $2$
Conductor $1920$
Sign $-0.184 + 0.982i$
Analytic cond. $15.3312$
Root an. cond. $3.91551$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + (1.76 − 1.37i)5-s + (0.159 − 0.159i)7-s + 9-s + (−1.60 − 1.60i)11-s − 4.36i·13-s + (1.76 − 1.37i)15-s + (−4.63 + 4.63i)17-s + (−3.97 − 3.97i)19-s + (0.159 − 0.159i)21-s + (−5.58 − 5.58i)23-s + (1.20 − 4.85i)25-s + 27-s + (6.25 − 6.25i)29-s − 1.69i·31-s + ⋯
L(s)  = 1  + 0.577·3-s + (0.787 − 0.615i)5-s + (0.0602 − 0.0602i)7-s + 0.333·9-s + (−0.485 − 0.485i)11-s − 1.21i·13-s + (0.454 − 0.355i)15-s + (−1.12 + 1.12i)17-s + (−0.912 − 0.912i)19-s + (0.0348 − 0.0348i)21-s + (−1.16 − 1.16i)23-s + (0.241 − 0.970i)25-s + 0.192·27-s + (1.16 − 1.16i)29-s − 0.304i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.184 + 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.184 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1920\)    =    \(2^{7} \cdot 3 \cdot 5\)
Sign: $-0.184 + 0.982i$
Analytic conductor: \(15.3312\)
Root analytic conductor: \(3.91551\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1920} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1920,\ (\ :1/2),\ -0.184 + 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.956414732\)
\(L(\frac12)\) \(\approx\) \(1.956414732\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
5 \( 1 + (-1.76 + 1.37i)T \)
good7 \( 1 + (-0.159 + 0.159i)T - 7iT^{2} \)
11 \( 1 + (1.60 + 1.60i)T + 11iT^{2} \)
13 \( 1 + 4.36iT - 13T^{2} \)
17 \( 1 + (4.63 - 4.63i)T - 17iT^{2} \)
19 \( 1 + (3.97 + 3.97i)T + 19iT^{2} \)
23 \( 1 + (5.58 + 5.58i)T + 23iT^{2} \)
29 \( 1 + (-6.25 + 6.25i)T - 29iT^{2} \)
31 \( 1 + 1.69iT - 31T^{2} \)
37 \( 1 - 0.609iT - 37T^{2} \)
41 \( 1 + 0.538iT - 41T^{2} \)
43 \( 1 - 0.592iT - 43T^{2} \)
47 \( 1 + (-4.85 - 4.85i)T + 47iT^{2} \)
53 \( 1 - 4.82T + 53T^{2} \)
59 \( 1 + (5.78 - 5.78i)T - 59iT^{2} \)
61 \( 1 + (1.65 + 1.65i)T + 61iT^{2} \)
67 \( 1 + 0.485iT - 67T^{2} \)
71 \( 1 - 6.86T + 71T^{2} \)
73 \( 1 + (0.160 - 0.160i)T - 73iT^{2} \)
79 \( 1 + 7.13T + 79T^{2} \)
83 \( 1 - 6.88T + 83T^{2} \)
89 \( 1 - 17.0T + 89T^{2} \)
97 \( 1 + (-9.64 + 9.64i)T - 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.685717323705707862251591437103, −8.473198910063088918741523137209, −7.67234021042630078796058506141, −6.30066432913366363325051446974, −6.02558721593459834169635205992, −4.75727457278354598203556935388, −4.16774977845325908235225900664, −2.74377866924992750579867619138, −2.13263552385949410110247430486, −0.60636701039157573298229746851, 1.82376750754717442576599772008, 2.33468484358730546389497298753, 3.49819115034836565605405439379, 4.48538943655673205485136234383, 5.38295385906155338538708019394, 6.49859029515984331416145234693, 6.94389491213567020976015106129, 7.83303315125428167892800991370, 8.821385041781061277236496379989, 9.352556410894495757818726488618

Graph of the $Z$-function along the critical line